The index associated to a symmetric, non-degenerate, and bilinear over a finite-dimensional vector space is a nonnegative integer defined by

where the set is defined to be

As a concrete example, a pair consisting of a smooth manifold with a symmetric tensor field is said to be a Lorentzian manifold if and only if and the index associated to the quadratic form satisfies for all (Sachs and Wu 1977). This particular definition succinctly conveys the fact that Lorentzian manifolds have indefinite metric tensors of signature (or equivalently ) without having to make precise any definitions related to metric signatures, quadratic form signatures, etc.

The above example also illustrates the deep connection between the index of a quadratic form and the notion of the index of a metric
tensor
defined on a smooth manifold . In particular, the index of a metric tensor is defined to be the quadratic form index associated to for any element . Because of this connection, indices are especially significant
to various fields: In some literature on differential and Riemannian geometry, for
example, the notion of an index is used as a main tool by which to *define*
a metric tensor (Sachs and Wu 1977).